Ancient Logic

Logic as a discipline starts with the transition from the more or less
unreflective use of logical methods and argument patterns to the
reflection on and inquiry into these methods and patterns and their
elements, including the syntax and semantics of sentences. In Greek
and Roman antiquity, discussions of some elements of logic and a focus
on methods of inference can be traced back to the late 5th
century BCE. The Sophists, and later Plato (early 4th c.)
displayed an interest in sentence analysis, truth, and fallacies, and
Eubulides of Miletus (mid-4th c.) is on record as the
inventor of both the Liar and the Sorites paradox. But logic as a
fully systematic discipline begins with Aristotle, who systematized
much of the logical inquiry of his predecessors. His main achievements
were his theory of the logical interrelation of affirmative and
negative existential and universal statements and, based on this
theory, his syllogistic, which can be interpreted as a system of
deductive inference. Aristotle's logic is known as term-logic, since
it is concerned with the logical relations between terms, such as
‘human being’, ‘animal’,
‘white’. It shares elements with both set theory and
predicate logic. Aristotle's successors in his school, the Peripatos,
notably Theophrastus and Eudemus, widened the scope of deductive
inference and improved some aspects of Aristotle's logic.

In the Hellenistic period, and apparently independent of Aristotle's
achievements, the logician Diodorus Cronus and his pupil Philo (see
the entry
Dialectical school)
worked out the beginnings of a logic that took propositions, rather
than terms, as its basic elements. They influenced the second major
theorist of logic in antiquity, the Stoic Chrysippus
(mid-3rd c.), whose main achievement is the development of
a propositional logic, crowned by a deductive system. Regarded by many
in antiquity as the greatest logician, he was innovative in a large
number of topics that are central to contemporary formal and
philosophical logic. The many close similarities between Chrysippus'
philosophical logic and that of Gottlob Frege are especially
striking. Chrysippus' Stoic successors systematized his logic, and
made some additions.

The development of logic from c. 100 BCE to c. 250 CE remains mostly
in the dark, but there can be no doubt that logic was one of the
topics regularly studied and researched. At some point Peripatetics
and Stoics began taking notice of each other's logical systems, and we
witness some conflation of both terminologies and theories.
Aristotelian syllogistic became known as ‘categorical
syllogistic’ and the Peripatetic adaptation of Stoic syllogistic
as ‘hypothetical syllogistic’. In the 2nd
century CE, Galen attempted to synthesize the two traditions; he also
professed to have introduced a third kind of syllogism, the
‘relational syllogism’, which apparently was meant to help
formalize mathematical reasoning. The attempt of some Middle
Platonists (1st c. BCE–2nd c. CE) to claim
a specifically Platonic logic failed, and in its stead, the
Neo-Platonists (3rd–6th c. CE) adopted a
scholasticized version of Aristotelian logic as their own. In the
monumental—if rarely creative—volumes of the Greek
commentators on Aristotle's logical works we find elements of Stoic
and later Peripatetic logic as well as Platonism, and ancient
mathematics and rhetoric. Much the same holds for the Latin logical
writings by Apuleius (2nd c. CE) and Boethius
(6th c. CE), which pave the way for Aristotelian logic,
thus supplemented, to enter the Medieval era.

Some of the Sophists classified types of sentences (logoi)
according to their force. So Protagoras (485–415 BCE), who
included wish, question, answer and command (Diels Kranz (DK) 80.A1,
Diogenes Laertius (D. L.) 9.53–4), and Alcidamas (pupil of
Gorgias, fl. 4th BCE), who distinguished assertion
(phasis), denial (apophasis), question and address
(prosagoreusis) (D. L. 9.54). Antisthenes
(mid-5th–mid-4th cent.) defined a sentence
as ‘that which indicates what a thing was or is’
(D. L. 6.3, DK 45) and stated that someone who says what is speaks
truly (DK49). Perhaps the earliest surviving passage on logic is found
in the Dissoi Logoi or Double Arguments (DK 90.4,
c. 400 BCE). It is evidence for a debate over truth and
falsehood. Opposed were the views (i) that truth is
a—temporal—property of sentences, and that a sentence is
true (when it is said), if and only if things are as the sentence says
they are when it is said, and false if they aren't; and (ii) that
truth is an atemporal property of what is said, and that what is said
is true if and only if the things are the case, false if they aren't
the case. These are rudimentary formulations of two alternative
correspondence theories of truth. The same passage displays awareness
of the fact that self-referential use of the truth-predicate can be
problematic—an insight also documented by the discovery of the
Liar paradox by Eubulides of Miletus (mid-4th c. BCE)
shortly thereafter.

Some Platonic dialogues contain passages whose topic is indubitably
logic. In the Sophist, Plato analyzes simple statements as
containing a verb (rhêma), which indicates action, and
a noun(onoma), which indicates the agent (Soph.
261e–262a). Anticipating the modern distinction of logical
types, he argues that neither a series of nouns nor a series of verbs
can combine into a statement (Soph. 262a–d). Plato also
divorces syntax (‘what is a statement?’) from semantics
(‘when is it true?’). Something (e.g. ‘Theaetetus is
sitting’) is a statement if it both succeeds in specifying a
subject and says something about this subject. Plato thus determines
subject and predicate as relational elements in a statement and
excludes as statements subject-predicate combinations containing empty
subject expressions. Something is a true statement if with reference
to its subject (Theaetetus) it says of what is (e.g. sitting) that it
is. Something is a false statement if with reference to its subject it
says of something other than what is (e.g. flying), that it is. Here
Plato produces a sketch of a deflationist theory of truth
(Soph. 262e–263d; cf. Crat. 385b). He also
distinguished negations from affirmations and took the negation
particle to have narrow scope: it negates the predicate, not the whole
sentence (Soph. 257b–c). There are many passages in
Plato where he struggles to explain certain logical relations:
for example his theory that things participate in Forms corresponds to
a rudimentary theory of predication; in the Sophist and
elsewhere he grapples with the class relations of exclusion, union and
co-extension; also with the difference between the ‘is’ of
predication (being) and the ‘is’ of identity (sameness);
and in Republic 4, 436bff., he anticipates the law of
non-contradiction. But his explications of these logical questions are
cast in metaphysical terms, and so can at most be regarded as
proto-logical.

Pre-Aristotelian evidence for reflection on argument forms and valid
inference are harder to come by. Both Zeno of Elea (born c. 490 BCE)
and Socrates (470–399) were famous for the ways in which they
refuted an opponent's view. Their methods display similarities with
reductio ad absurdum, but neither of them seems to have
theorized about their logical procedures. Zeno produced arguments
(logoi) that manifest variations of the pattern ‘this
(i.e. the opponent's view) only if that. But that is impossible. So
this is impossible’. Socratic refutation was an exchange of
questions and answers in which the opponents would be led, on the
basis of their answers, to a conclusion incompatible with their
original claim. Plato institutionalized such disputations into
structured, rule-governed verbal contests that became known as
dialectical argument. The development of a basic logical vocabulary
for such contests indicates some reflection upon the patterns of
argumentation.

The 5th and early to mid-4th centuries BCE also
see great interest in fallacies and logical paradoxes. Besides the
Liar, Eubulides is said to have been the originator of several other
logical paradoxes, including the Sorites. Plato's Euthydemus
contains a large collection of contemporary fallacies. In attempts to
solve such logical puzzles, a logical terminology develops here, too,
and the focus on the difference between valid and invalid arguments
sets the scene for the search for a criterion of valid
inference. Finally, it is possible that the shaping of deduction and
proof in Greek mathematics that begins in the later 5th
century BCE served as an inspiration for Aristotle's syllogistic.

(For a more detailed account see the entry on
Aristotle's Logic
in this encyclopedia.) Aristotle is the first great logician in the
history of logic. His logic was taught by and large without rival from
the 4th to the 19th centuries CE. Aristotle's
logical works were collected and put in a systematic order by later
Peripatetics, who entitled them the Organon or
‘tool’, because they considered logic not a part but
rather an instrument of philosophy. The Organon contains, in
traditional order, the Categories, De
Interpretatione, Prior Analytics, Posterior
Analytics, Topics and Sophistical Refutations.
In addition, Metaphysics Γ is a logical treatise that
discusses the principle of non-contradiction, and some further logical
insights are found scattered throughout Aristotle's other works, such
as the Poetics, Rhetoric, De Anima,
Metaphysics Δ and Θ, and some of the biological
works. Some parts of the Categories and Posterior
Analytics would today be regarded as metaphysics, epistemology or
philosophy of science rather than logic. The traditional arrangement
of works in the Organon is neither chronological nor
Aristotle's own. The original chronology cannot be fully recovered
since Aristotle seems often to have inserted supplements into earlier
writings at a later time. However, by using logical advances as a
criterion, we can conjecture that most of the Topics,
Sophistical Refutations, Categories and
Metaphysics Γ predate the De Interpretatione,
which in turn predates the Prior Analytics and parts of the
Posterior Analytics.

The Topics provide a manual for participants in the contests
of dialectical argument as instituted in the Academy by Plato. Books
2–7 provide general procedures or rules (topoi) about
how to find an argument to establish or refute a given thesis. The
descriptions of these procedures—some of which are so general
that they resemble logical laws—clearly presuppose a notion of
logical form, and Aristotle's Topics may thus count as the
earliest surviving logical treatise. The Sophistical
Refutations are the first systematic classification of fallacies,
sorted by what logical flaw each type manifests (e.g. equivocation,
begging the question, affirming the consequent, secundum
quid) and how to expose them.

Aristotle distinguishes things that have sentential unity through a
combination of expressions (‘a horse runs’) from those
that do not (‘horse’, ‘runs’); the latter are
dealt with in the Categories (the title really means
‘predications’[1]).
They have no truth-value and signify one of the following: substance
(ousia), quantity (poson), quality (poion),
relation (pros ti), location (pou), time
(pote), position (keisthai), possession
(echein), doing (poiein) and undergoing
(paschein). It is unclear whether Aristotle considers this
classification to be one of linguistic expressions that can be
predicated of something else; or of kinds of predication; or of
highest genera. In Topics 1 Aristotle distinguishes four
relationships a predicate may have to the subject: it may give its
definition, genus, unique property, or accidental property. These are
known as predicables.

When writing the De Interpretatione, Aristotle had worked out
the following theory of simple sentences: a (declarative) sentence
(apophantikos logos) or declaration (apophansis) is
delimited from other pieces of discourse like prayer, command and
question by its having a truth-value. The truth-bearers that feature
in Aristotle's logic are thus linguistic items. They are spoken
sentences that directly signify thoughts (shared by all humans) and
through these, indirectly, things. Written sentences in turn signify
spoken ones. (Simple) sentences are constructed from two signifying
expressions which stand in subject-predicate relation to each other: a
name and a verb (‘Callias walks’) or two names connected
by the copula ‘is’, which co-signifies the connection
(‘Pleasure is good’) (Int. 3). Names are either
singular terms or common nouns (An. Pr. I 27). Both can be
empty (Cat. 10, Int. 1). Singular terms can only
take subject position. Verbs co-signify time. A name-verb sentence can
be rephrased with the copula (‘Callias is (a) walking
(thing)’) (Int. 12). As to their quality, a
(declarative) sentence is either an affirmation or a negation,
depending on whether it affirms or negates its predicate of its
subject. The negation particle in a negation has wide scope
(Cat. 10). Aristotle defined truth separately for
affirmations and negations: An affirmation is true if it says of that
which is that it is; a negation is true if it says of that which is
not that it is not (Met. Γ.7 1011b25ff). These
formulations, or in any case their Greek counterparts, can be
interpreted as expressing either a correspondence or a deflationist
conception of truth. Either way, truth is a property that belongs to a
sentence at a given time. As to their quantity, sentences are
singular, universal, particular or indefinite. Thus Aristotle obtains
eight types of sentences, which are later dubbed ‘categorical
sentences’. The following are examples, paired by quality:

Singular:

Callias is just.

Callias is not just.

Universal:

Every human is just.

No human is just.

Particular:

Some human is just.

Some human is not just.

Indefinite:

(A) human is just.

(A) human is not just.

Universal and particular sentences contain a quantifier and both
universal and particular affirmatives were taken to have existential
import. (See entry
The Traditional Square of Opposition).
The logical
status of the indefinites is ambiguous and controversial
(Int. 6–7).

Aristotle distinguishes between two types of sentential opposition:
contraries and contradictories. A contradictory pair of sentences (an
antiphasis) consists of an affirmation and its negation (i.e.
the negation that negates of the subject what the affirmation affirms
of it). Aristotle assumes that—normally—one of these must
be true, the other false. Contrary sentences are such that they cannot
both be true. The contradictory of a universal affirmative is the
corresponding particular negative; that of the universal negative the
corresponding particular affirmative. A universal affirmative and its
corresponding universal negative are contraries. Aristotle thus has
captured the basic logical relations between monadic quantifiers
(Int. 7).

Since Aristotle regards tense as part of the truth-bearer (as opposed
to merely a grammatical feature), he detects a problem regarding
future tense sentences about contingent matters: Does the principle
that of an affirmation and its negation one must be false, the other
true, apply to these? What, for example, is the truth-value now of the
sentence ‘There will be a sea-battle tomorrow’? Aristotle
may have suggested that the sentence has no truth-value now, and that
bivalence thus does not hold—despite the fact that it is
necessary for there either to be or not to be a sea-battle tomorrow,
so that the principle of excluded middle is preserved (Int.
9).

Aristotle's non-modal syllogistic (Prior Analytics A
1–7) is the pinnacle of his logic. Aristotle defines a syllogism
as ‘an argument (logos) in which, certain things having
been laid down, something different from what has been laid down
follows of necessity because these things are so’. This
definition appears to require (i) that a syllogism consists of at
least two premises and a conclusion, (ii) that the conclusion follows
of necessity from the premises (so that all syllogisms are
valid arguments), and (iii) that the conclusion differs from
the premises. Aristotle's syllogistic covers only a small part of all
arguments that satisfy these conditions.

Aristotle restricts and regiments the types of categorical sentence
that may feature in a syllogism. The admissible truth-bearers are now
defined as each containing two different terms (horoi)
conjoined by the copula, of which one (the predicate term) is said of
the other (the subject term) either affirmatively or negatively.
Aristotle never comes clear on the question whether terms are things
(e.g., non-empty classes) or linguistic expressions for these things.
Only universal and particular sentences are discussed. Singular
sentences seem to be excluded and indefinite sentences are mostly
ignored. At An. Pr. A 7 Aristotle mentions that by substituting an
indefinite premise for a particular, one obtains a syllogism of
the same kind.

Another innovation in the syllogistic is Aristotle's use of letters in
place of terms. The letters may originally have served simply as
abbreviations for terms (e.g. An. Post. A 13); but in the
syllogistic they seem mostly to have the function either of schematic
term letters or of term variables with universal quantifiers assumed
but not stated. Where he uses letters, Aristotle tends to express the
four types of categorical sentences in the following way (with common
later abbreviations in parentheses):

‘A holds of (lit., belongs to) every
B’

(AaB)

‘A holds of no B’

(AeB)

‘A holds of some B’

(AiB)

‘A does not hold of some B’

(AoB)

Instead of ‘holds’ he also uses ‘is
predicated’.

All basic syllogisms consist of three categorical sentences, in which
the two premises share exactly one term, called the middle term, and
the conclusion contains the other two terms, sometimes called the
extremes. Based on the position of the middle term, Aristotle
classified all possible premise combinations into three figures
(schêmata): the first figure has the middle term (B)
as subject in the first premise and predicated in the second; the
second figure has it predicated in both premises, the third has it as
subject in both premises:

I

II

III

A holds of B

B holds of A

A holds of B

B holds of C

B holds of C

C holds of B

A is also called the major term, C the minor
term. Each figure can further be classified according to whether or
not both premises are universal. Aristotle went systematically through
the fifty-eight possible premise combinations and showed that fourteen
have a conclusion following of necessity from them, i.e. are
syllogisms. His procedure was this: He assumed that the syllogisms of
the first figure are complete and not in need of proof, since they are
evident. By contrast, the syllogisms of the second and third figures
are incomplete and in need of proof. He proves them by reducing them
to syllogisms of the first figure and thereby ‘completing’
them. For this he makes use of three methods:

(i) conversion (antistrophê): a categorical sentence is
converted by interchanging its terms. Aristotle recognizes and
establishes three conversion rules: ‘from AeB
infer BeA’; ‘from AiB
infer BiA’ and ‘from
AaB infer BiA’. All but two second and third figure syllogisms can be proved by premise
conversion.

(ii) reductio ad impossibile (apagôgê):
the remaining two are proved by reduction to the impossible, where the
contradictory of an assumed conclusion together with one of the
premises is used to deduce by a first figure syllogism a conclusion
that is incompatible with the other premise. Using the semantic
relations between opposites established earlier the assumed conclusion
is thus established.

(iii) exposition or setting-out (ekthesis): this method,
which Aristotle uses in addition to (i) and (ii), involves choosing
or ‘setting out’ some additional term, say D,
that falls in the non-empty intersection delimited by two premises,
say AxB and AxC,
and using D to justify the inference from the premises to a
particular conclusion, BxC. It is debated
whether ‘D’ represents a singular or a general
term and whether exposition constitutes proof.

For each of the thirty-four premise combinations that allow no
conclusion Aristotle proves by counterexample that they allow no
conclusion. As his overall result, he acknowledges four first figure
syllogisms (later named Barbara, Celarent, Darii, Ferio), four second
figure syllogisms (Camestres, Cesare, Festino, Baroco) and six third
figure syllogisms (Darapti, Felapton, Disamis, Datisi, Bocardo,
Ferison); these were later called the modes or moods of the figures.
(The names are mnemonics: e.g. each vowel, or the first three in cases
where the name has more than three, indicates in order whether the
first and second premises and the conclusion were sentences of type
a, e, i or o.) Aristotle
implicitly recognized that by using the conversion rules on the
conclusions we obtain eight further syllogisms (An. Pr.
53a3–14), and that of the premise combinations rejected as
non-syllogistic, some (five, in fact) will yield a conclusion in which
the minor term is predicated of the major (An. Pr.
29a19–27). Moreover, in the Topics Aristotle accepted
the rules ‘from AaB infer
AiB’ and ‘from AeB
infer AoB’. By using these on the conclusions
five further syllogisms could be proved, though Aristotle did not
mention this.

Going beyond his basic syllogistic, Aristotle reduced the
3rd and 4th first figure syllogisms to second
figure syllogisms, thus de facto reducing all syllogisms to
Barbara and Celarent; and later on in the Prior Analytics he
invokes a type of cut-rule by which a multi-premise syllogism can be
reduced to two or more basic syllogisms. From a modern perspective,
Aristotle's system can be understood as a sequent logic in the style
of natural deduction and as a fragment of first-order logic. It has
been shown to be sound and complete if one interprets the relations
expressed by the categorical sentences set-theoretically as a system
of non-empty classes as follows: AaB is true if and
only if the class A contains the class
B. AeB is true if and only if the classes
A and B are disjoint. AiB is true
if and only if the classes A and B are not
disjoint. AoB is true if and only if the class
A does not contain the class B. It is generally
agreed, though, that Aristotle's syllogistic is a kind of relevance
logic rather than classical. The vexing textual question what exactly
Aristotle meant by ‘syllogisms’ has received several rival
interpretations, including one that they are a certain type of
conditional propositional form. Most plausibly, perhaps, Aristotle's
complete and incomplete syllogisms taken together should be
understood as formally valid premise-conclusion arguments; and his
complete and completed syllogisms taken together as (sound)
deductions.

Aristotle is also the originator of modal logic. In addition to
quality (as affirmation or negation) and quantity (as singular,
universal, particular, or indefinite), he takes categorical sentences
to have a mode; this consists of the fact that the predicate is said
to hold of the subject either actually or necessarily or possibly or
contingently or impossibly. The latter four are expressed by modal
operators that modify the predicate, e.g. ‘It is possible for
A to hold of some B’; ‘A
necessarily holds of every B’.

In De Interpretatione 12–13, Aristotle (i) concludes
that modal operators modify the whole predicate (or the copula, as he
puts it), not just the predicate term of a sentence. (ii) He states
the logical relations that hold between modal operators, such as that
‘it is not possible for A not to hold of
B’ implies ‘it is necessary for A to
hold of B’. (iii) He investigates what the
contradictories of modalized sentences are, and decides that they are
obtained by placing the negator in front of the modal operator. (iv)
He equates the expressions ‘possible’ and
‘contingent’, but wavers between a one-sided
interpretation (where necessity implies possibility) and a two-sided
interpretation (where possibility implies non-necessity).

Aristotle develops his modal syllogistic in Prior Analytics
1.8–22. He settles on two-sided possibility (contingency) and
tests for syllogismhood all possible combinations of premise pairs of
sentences with necessity (N), contingency (C) or no (U) modal
operator: NN, CC, NU/UN, CU/UC and NC/CN. Syllogisms with the last
three types of premise combinations are called mixed modal
syllogisms. Apart from the NN category, which mirrors unmodalized
syllogisms, all categories contain dubious cases. For instance,
Aristotle accepts:

A necessarily holds of all B.B holds of all C.
Therefore A necessarily holds of all C.

This and other problematic cases were already disputed in antiquity,
and more recently have sparked a host of complex formalized
reconstructions of Aristotle's modal syllogistic. As Aristotle's
theory is conceivably internally inconsistent, the formal models that
have been suggested may all be unsuccessful.

Aristotle's pupil and successor Theophrastus of Eresus
(c. 371–c. 287 BCE) wrote more logical treatises than his
teacher, with a large overlap in topics. Eudemus of Rhodes (later
4th cent. BCE) wrote books entitled Categories,
Analytics and On Speech. Of all these works only a
number of fragments and later testimonies survive, mostly in
commentators on Aristotle. Theophrastus and Eudemus simplified some aspects of
Aristotle's logic, and developed others where Aristotle left us only
hints.

The two Peripatetics seem to have redefined Aristotle's first figure,
so that it includes every syllogism in which the middle term is
subject of one premise and predicate of the other. In this way, five
types of non-modal syllogisms only intimated by Aristotle later in his
Prior Analytics (Baralipton, Celantes, Dabitis, Fapesmo and
Frisesomorum) are included, but Aristotle's criterion that first
figure syllogisms are evident is given up (Theophrastus fr. 91,
Fortenbaugh). Theophrastus and Eudemus also improved Aristotle's modal
theory. Theophrastus replaced Aristotle's two-sided contingency with
one-sided possibility, so that possibility no longer entails
non-necessity. Both recognized that the problematic universal negative
(‘A possibly holds of no B’) is simply
convertible (Theophrastus fr. 102A Fortenbaugh). Moreover, they
introduced the principle that in mixed modal syllogisms the conclusion
always has the same modal character as the weaker of the premises
(Theophrastus frs. 106 and 107 Fortenbaugh), where possibility is
weaker than actuality, and actuality than necessity. In this way
Aristotle's modal syllogistic is notably simplified and many
unsatisfactory theses, like the one mentioned above (that from
‘Necessarily AaB’ and
‘BaC’ one can infer ‘Necessarily
AaC’) disappear.

Theophrastus introduced the so-called prosleptic premises and
syllogisms (Theophrastus fr. 110 Fortenbaugh). A prosleptic premise is
of the form:

For all X, if Φ(X), then Ψ(X)

where Φ(X) and Ψ(X) stand for categorical
sentences in which the variable X occurs in place of one of
the terms. For example:

(1) A [holds] of all of that of all of
which B [holds].

(2) A [holds] of none of that which
[holds] of all B.

Theophrastus considered such premises to contain three terms, two of
which are definite (A, B), one indefinite
(‘that’, or the bound variable X). We can
represent (1) and (2) as

∀X (BaX →
AaX)

∀X (XaB →
AeX)

Prosleptic syllogisms then come about as follows: They are composed of
a prosleptic premise and the categorical premise obtained by
instantiating a term (C) in the antecedent ‘open
categorical sentence’ as premises, and the categorical sentences
one obtains by putting in the same term (C) in the consequent
‘open categorical sentence’ as conclusion. For
example:

A [holds] of all of that of all of which B
[holds].B holds of all C.
Therefore, A holds of all C.

Theophrastus distinguished three figures of these syllogisms,
depending on the position of the indefinite term (also called
‘middle term’) in the prosleptic premise; for example (1)
produces a third figure syllogism, (2) a first figure syllogism. The
number of prosleptic syllogisms was presumably equal to that of types
of prosleptic sentences: with Theophrastus' concept of the first
figure these would be sixty-four (i.e. 32 + 16 + 16). Theophrastus
held that certain prosleptic premises were equivalent to certain
categorical sentences, e.g. (1) to ‘A is predicated of
all B’. However, for many, including (2), no such
equivalent can be found, and prosleptic syllogisms thus increased the
inferential power of Peripatetic logic.

Theophrastus and Eudemus considered complex premises which they called
‘hypothetical premises’ and which had one of the following
two (or similar) forms:

If something is F, it is G

Either something is F or it is G (with
exclusive ‘or’)

They developed arguments with them which they called ‘mixed from
a hypothetical premise and a probative premise’ (Theophrastus
fr. 112A Fortenbaugh). These arguments were inspired by Aristotle's
syllogisms ‘from a hypothesis’ (An. Pr. 1.44);
they were forerunners of modus ponens and modus
tollens and had the following forms (Theophrastus frs. 111 and
112 Fortenbaugh), employing the exclusive ‘or’:

If something is F, it is G.a is F.
Therefore, a is G.

If something is F, it is G.a is not G.
Therefore, a is not F.

Either something is F or it is G.a is F.
Therefore, a is not G.

Either something is F or it is G.a is not F.
Therefore, a is G.

Theophrastus also recognized that the connective particle
‘or’ can be inclusive (Theophrastus fr. 82A Fortenbaugh);
and he considered relative quantified sentences such as those
containing ‘more’, ‘fewer’, and ‘the
same’ (Theophrastus fr. 89 Fortenbaugh), and seems to have
discussed syllogisms built from such sentences, again following up
upon what Aristotle said about syllogisms from a hypothesis
(Theophrastus fr. 111E Fortenbaugh).

Theophrastus is further credited with the invention of a system of the
later so-called ‘wholly hypothetical syllogisms’
(Theophrastus fr. 113 Fortenbaugh). These syllogisms were originally
abbreviated term-logical arguments of the kind

and at least some of them were regarded as reducible to Aristotle's
categorical syllogisms, presumably by way of the equivalences to
‘Every A is B’, etc. In parallel to
Aristotle's syllogistic, Theophrastus distinguished three figures;
each had sixteen modes. The first eight modes of the first figure are
obtained by going through all permutations with ‘not
X’ instead of ‘X’ (with X
for A, B, C); the second eight modes are
obtained by using a rule of contraposition on the conclusion:

(CR) From ‘if X, Y’ infer
‘if the contradictory of Y then the contradictory of
X’

The sixteen modes of the second figure were obtained by using (CR) on
the schema of the first premise of the first figure arguments,
e.g.

Theophrastus claimed that all second and third figure syllogisms could
be reduced to first figure syllogisms. If Alexander of Aphrodisias
(2nd c. CE Peripatetic) reports faithfully, any use of (CR)
which transforms a syllogism into a first figure syllogism was such a
reduction. The large number of modes and reductions can be explained
by the fact that Theophrastus did not have the logical means for
substituting negative for positive components in an argument. In later
antiquity, after some intermediate stages, and possibly under Stoic
influence, the wholly hypothetical syllogisms were interpreted as
propositional-logical arguments of the kind

In the later 4th to mid 3rd centuries BCE,
contemporary with Theophrastus and Eudemus, a loosely connected group
of philosophers, sometimes referred to as dialecticians (see entry
‘Dialectical School’) and possibly influenced by
Eubulides, conceived of logic as a logic of propositions. Their best
known exponents were Diodorus Cronus and his pupil Philo (sometimes
called ‘Philo of Megara’). Although no writings of theirs
are preserved, there are a number of later reports of their
doctrines. They each made groundbreaking contributions to the
development of propositional logic, in particular to the theories of
conditionals and modalities.

A conditional (sunêmmenon) was considered a
non-simple proposition composed of two propositions and the connecting
particle ‘if’. Philo, who may be credited with introducing
truth-functionality into logic, provided the following criterion for
their truth: A conditional is false when and onlywhen its antecedent is true and its consequent is false, and
it is true in the three remaining truth-value combinations. The
Philonian conditional resembles material implication, except
that—since propositions were conceived of as functions of time
that can have different truth-values at different times—it may
change its truth-value over time. For Diodorus, a conditional
proposition is true if it neither was nor is possible that its
antecedent is true and its consequent false. The temporal elements in
this account suggest that the possibility of a truth-value change in
Philo's conditionals was meant to be improved on. With his own modal
notions (see below) applied, a conditional is Diodorean-true now if
and only if it is Philonian-true at all times. Diodorus' conditional
is thus reminiscent of strict implication. Philo's and Diodorus'
conceptions of conditionals lead to variants of the
‘paradoxes’ of material and strict implication—a
fact the ancients were aware of (Sextus Empiricus [S. E.] M.
8.109–117).

Philo and Diodorus each considered the four modalities possibility,
impossibility, necessity and non-necessity. These were conceived of as
modal properties or modal values of propositions, not as modal
operators. Philo defined them as follows: ‘Possible is that
which is capable of being true by the proposition's own nature
… necessary is that which is true, and which, as far as it is
in itself, is not capable of being false. Non-necessary is that which
as far as it is in itself, is capable of being false, and impossible
is that which by its own nature is not capable of being true.’
Diodorus' definitions were these: ‘Possible is that which either
is or will be [true]; impossible that which is false and will
not be true; necessary that which is true and will not be false;
non-necessary that which either is false already or will be
false.’ Both sets of definitions satisfy the following standard
requirements of modal logic: (i) necessity entails truth and truth
entails possibility; (ii) possibility and impossibility are
contradictories, and so are necessity and non-necessity; (iii)
necessity and possibility are interdefinable; (iv) every proposition
is either necessary or impossible or both possible and
non-necessary. Philo's definitions appear to introduce mere conceptual
modalities, whereas with Diodorus' definitions, some propositions may
change their modal value (Boeth. In Arist. De Int., sec. ed.,
234–235 Meiser).

Diodorus' definition of possibility rules out future contingents and
implies the counterintuitive thesis that only the actual is
possible. Diodorus tried to prove this claim with his famous Master
Argument, which sets out to show the incompatibility of (i)
‘every past truth is necessary’, (ii) ‘the
impossible does not follow from the possible’, and (iii)
‘something is possible which neither is nor will be true’
(Epict. Diss. II.19). The argument has not survived, but
various reconstructions have been suggested. Some affinity with the
arguments for logical determinism in Aristotle's De
Interpretatione 9 is likely.

On the topic of ambiguity, Diodorus held that no linguistic expression
is ambiguous. He supported this dictum by a theory of meaning based on
speaker intention. Speakers generally intend to say only one thing
when they speak. What is said when they speak is what they intend to
say. Any discrepancy between speaker intention and listener decoding
has its cause in the obscurity of what was said, not its ambiguity
(Aulus Gellius 11.12.2–3).

The founder of the Stoa, Zeno of Citium (335–263 BCE), studied
with Diodorus. His successor Cleanthes (331–232) tried to solve
the Master Argument by denying that every past truth is necessary and
wrote books—now lost—on paradoxes, dialectics, argument
modes and predicates. Both philosophers considered knowledge of logic
as a virtue and held it in high esteem, but they seem not to have been
creative logicians. By contrast, Cleanthes' successor Chrysippus of
Soli (c. 280–207) is without doubt the second great logician in
the history of logic. It was said of him that if the gods used any
logic, it would be that of Chrysippus (D. L. 7.180), and his reputation
as a brilliant logician is amply attested. Chrysippus wrote over 300
books on logic, on virtually every topic logic today concerns itself
with, including speech act theory, sentence analysis, singular and
plural expressions, types of predicates, indexicals, existential
propositions, sentential connectives, negations, disjunctions,
conditionals, logical consequence, valid argument forms, theory of
deduction, propositional logic, modal logic, tense logic, epistemic
logic, logic of suppositions, logic of imperatives, ambiguity and
logical paradoxes, in particular the Liar and the Sorites
(D. L. 7.189–199). Of all these, only two badly damaged papyri
have survived, luckily supplemented by a considerable number of
fragments and testimonies in later texts, in particular in Diogenes
Laertius (D. L.) book 7, sections 55–83, and Sextus Empiricus
Outlines of Pyrrhonism (S. E. PH) book 2 and
Against the Mathematicians (S. E. M) book
8. Chrysippus' successors, including Diogenes of Babylon (c.
240–152) and Antipater of Tarsus (2nd cent. BCE),
appear to have systematized and simplified some of his ideas, but
their original contributions to logic seem small. Many testimonies of
Stoic logic do not name any particular Stoic. Hence the following
paragraphs simply talk about ‘the Stoics’ in general; but
we can be confident that a large part of what has survived goes back
to Chrysippus.

The subject matter of Stoic logic is the so-called sayables
(lekta): they are the underlying meanings in everything we
say and think, but—like Frege's 'senses'—also subsist
independently of us. They are distinguished from spoken and written
linguistic expressions: what we utter are those expressions,
but what we say are the sayables (D. L. 7.57). There are
complete and deficient sayables. Deficient sayables, if said, make the
hearer feel prompted to ask for a completion; e.g. when someone says
‘writes’ we enquire ‘who?’. Complete sayables,
if said, do not make the hearer ask for a completion (D. L.7.63). They
include assertibles (the Stoic equivalent of propositions),
imperativals, interrogatives, inquiries, exclamatives, hypotheses or
suppositions, stipulations, oaths, curses and more. The accounts of
the different complete sayables all had the general form ‘a
so-and-so sayable is one in saying which we perform an act of
such-and-such’. For instance: ‘an imperatival sayable is
one in saying which we issue a command’, ‘an interrogative
sayable is one in saying which we ask a question’, ‘a
declaratory sayable (i.e. an assertible) is one in saying which we
make an assertion’. Thus, according to the Stoics, each time we
say a complete sayable, we perform three different acts: we utter a
linguistic expression; we say the sayable; and we perform a
speech-act. Chrysippus was aware of the use-mention distinction
(D. L. 7.187). He seems to have held that every denoting expression is
ambiguous in that it denotes both its denotation and itself (Galen,
On ling. soph. 4; Aulus Gellius 11.12.1). Thus the
expression ‘a wagon’ would denote both a wagon and the
expression ‘a
wagon’.[2]

Assertibles (axiômata) differ from all other complete
sayables in their having a truth-value: at any one time they are
either true or false. Truth is temporal and assertibles may change
their truth-value. The Stoic principle of bivalence is hence
temporalized, too. Truth is introduced by example: the assertible
‘it is day’ is true when it is day, and at all
other times false (D. L. 7.65). This suggests some kind of deflationist
view of truth, as does the fact that the Stoics identify true
assertibles with facts, but define false assertibles simply as the
contradictories of true ones (S. E. M 8.85).

Assertibles are simple or non-simple. A simple predicative
assertible like ‘Dion is walking’ is generated from the
predicate ‘is walking’, which is a deficient assertible
since it elicits the question ‘who?’, together with a
nominative case (Dion's individual quality or the correlated sayable),
which the assertible presents as falling under the predicate
(D. L. 7.63 and 70). There is thus no interchangeability of predicate
and subject terms as in Aristotle; rather, predicates—but not
the things that fall under them—are defined as deficient, and
thus resemble propositional functions. It seems that whereas some
Stoics took the—Fregean—approach that singular terms had
correlated sayables, others anticipated the notion of direct
reference. Concerning indexicals, the Stoics took a simple
definite assertible like ‘this one is walking’ to
be true when the person pointed at by the speaker is walking (S. E.
M 100). When the thing pointed at ceases to be, so does the
assertible, though the sentence used to express it remains
(Alex. Aphr. An. Pr. 177–8). A simple
indefinite assertible like ‘someone is walking’
is said to be true when a corresponding definite assertible is true
(S. E. M 98). Aristotelian universal affirmatives
(‘Every A is B’) were to be rephrased as
conditionals: ‘If something is A, it is
B’ (S. E. M 9.8–11). Negations of simple
assertibles are themselves simple assertibles. The Stoic negation of
‘Dion is walking’ is ‘(It is) not (the case that)
Dion is walking’, and not ‘Dion is not walking’. The
latter is analyzed in a Russellian manner as ‘Both Dion exists
and not: Dion is walking’ (Alex. Aphr. An. Pr.
402). There are present tense, past tense and future tense
assertibles. The—temporalized—principle of bivalence
holds for them all. The past tense assertible ‘Dion
walked’ is true when there is at least one past time at which
‘Dion is walking’ was true.

Thus the Stoics concerned themselves with several issues we would
place under the heading of predicate logic; but their main achievement
was the development of a propositional logic, i.e. of a system of
deduction in which the smallest substantial unanalyzed expressions are
propositions, or rather, assertibles.

The Stoics defined negations as assertibles that consist of a negative
particle and an assertible controlled by this particle (S. E.
M8.103). Similarly, non-simple assertibles were defined as
assertibles that either consist of more than one assertible or of one
assertible taken more than once (D. L. 7.68–9) and that are
controlled by a connective particle. Both definitions can be
understood as being recursive and allow for assertibles of
indeterminate complexity. Three types of non-simple assertibles
feature in Stoic syllogistic. Conjunctions are non-simple assertibles
put together by the conjunctive connective ‘both … and
…’. They have two
conjuncts.[3]
Disjunctions are non-simple assertibles put together by the
disjunctive connective ‘either … or … or
…’. They have two or more disjuncts, all on a
par. Conditionals are non-simple assertibles formed with the
connective ‘if …, …’; they consist of
antecedent and consequent (D. L. 7.71–2). What type of assertible
an assertible is, is determined by the connective or logical particle
that controls it, i.e. that has the largest scope. ‘Both not
p and q’ is a conjunction, ‘Not both
p and q’ a negation. Stoic language
regimentation asks that sentences expressing assertibles always start
with the logical particle or expression characteristic for the
assertible. Thus, the Stoics invented an implicit bracketing device
similar to that used in Łukasiewicz' Polish notation.

Stoic negations and conjunctions are truth-functional. Stoic (or at
least Chrysippean) conditionals are true when the contradictory of the
consequent is incompatible with its antecedent (D. L. 7.73). Two
assertibles are contradictories of each other if one is the negation
of the other (D. L. 7.73); that is, when one exceeds the other by
a—pre-fixed—negation particle (S. E. M 8.89). The
truth-functional Philonian conditional was expressed as a negation of
a conjunction: that is, not as ‘if p, q’
but as ‘not both p and not q’. Stoic
disjunction is exclusive and non-truth-functional. It is true when
necessarily precisely one of its disjuncts is true. Later Stoics
introduced a non-truth-functional inclusive disjunction (Aulus
Gellius, N. A. 16.8.13–14).

Like Philo and Diodorus, Chrysippus distinguished four modalities and
considered them modal values of propositions rather than modal
operators; they satisfy the same standard requirements of modal logic.
Chrysippus' definitions are (D. L. 7.75): An assertible is possible
when it is both capable of being true and not hindered by external
things from being true. An assertible is impossible when it is
[either] not capable of being true [or is capable of being
true, but hindered by external things from being true]. An
assertible is necessary when, being true, it either is not capable of
being false or is capable of being false, but hindered by external
things from being false. An assertible is non-necessary when it is
both capable of being false and not hindered by external things
[from being false]. Chrysippus' modal notions differ from
Diodorus' in that they allow for future contingents and from Philo's
in that they go beyond mere conceptual possibility.

Arguments are—normally—compounds of assertibles. They are
defined as a system of at least two premises and a conclusion
(D. L. 7.45). Syntactically, every premise but the first is introduced
by ‘now’ or ‘but’, and the conclusion by
‘therefore’. An argument is valid if the (Chrysippean)
conditional formed with the conjunction of its premises as antecedent
and its conclusion as consequent is correct (S. E. PH 2.137;
D. L. 7.77). An argument is ‘sound’ (literally:
‘true’), when in addition to being valid it has true
premises. The Stoics defined so-called argument modes as a sort of
schema of an argument (D. L. 7.76). The mode of an argument differs from
the argument itself by having ordinal numbers taking the place of
assertibles. The mode of the argument

If it is day, it is light.
But it is not the case that it is light.
Therefore it is not the case that it is day.

is

If the 1st, the 2nd.
But not: the 2nd.
Therefore not: the 1st.

The modes functioned first as abbreviations of arguments that brought
out their logically relevant form; and second, it seems, as
representatives of the form of a class of arguments.

In terms of contemporary logic, Stoic syllogistic is best understood
as a substructural backwards-working Gentzen-style natural-deduction
system that consists of five kinds of axiomatic arguments (the
indemonstrables) and four inference rules, called themata. An
argument is a syllogism precisely if it either is an indemonstrable or
can be reduced to one by means of the themata
(D. L. 7.78). Thus syllogisms are certain kinds of formally valid
arguments. The Stoics explicitly acknowledged that there are valid
arguments that are not syllogisms; but assumed that these could be
somehow transformed into syllogisms.

All basic indemonstrables consist of a non-simple assertible as
leading premiss and a simple assertible as co-assumption, and have
another simple assertible as conclusion. They were defined by five
standardized meta-linguistic descriptions of the forms of the
arguments (S. E. M 8.224–5; D. L. 7.80–1):

A first indemonstrable is an argument that concludes from a
conditional and its antecedent the consequent <of the
conditional>

A second indemonstrable is an argument that concludes from a
conditional and the contradictory of the consequent the contradictory
of the antecedent <of the conditional>.

A third indemonstrable is an argument that concludes from the
negation of a conjunction and one of the conjuncts the contradictory
of the other conjunct.

A fourth indemonstrable is an argument that concludes from a
disjunction and one of the disjuncts the contradictory of the other
disjunct.

A fifth indemonstrable is an argument that concludes from a
disjunction and the contradictory of one of its disjuncts the other
disjunct.

Whether an argument is an indemonstrable can be tested by comparing
it with these meta-linguistic descriptions. For instance,

If it is day, it is not the case that it is night.
But it is night.
Therefore it is not the case that it is day.

comes out as a second indemonstrable, and

If five is a number, then either five is odd or five is even.
But five is a number.
Therefore either five is odd or five is even.

as a first indemonstrable. For testing, a suitable mode of an argument
can also be used as a stand-in. A mode is syllogistic, if a
corresponding argument with the same form is a syllogism (because of
that form). However in Stoic logic there are no five modes that can be
used as inference schemata that represent the five types of
indemonstrables. For example, the following are two of the many modes
of fourth indemonstrables:

Either the 1st or the 2nd.
But the 2nd.
Therefore not the 1st.

Either the 1st or not the 2nd.
But the 1st.
Therefore the 2nd.

Although both are covered by the meta-linguistic description, neither
could be singled out as the mode of the fourth
indemonstrables: If we disregard complex arguments, there are
thirty-two modes corresponding to the five meta-linguistic
descriptions; the latter thus prove noticeably more economical. The
almost universal assumption among historians of logic that the Stoics
represented their five (types of) indemonstrables by five modes is
false and not supported by textual
evidence.[4]

Of the four themata, only the first and third are extant.
They, too, were meta-linguistically formulated. The first
thema, in its basic form, was:

When from two [assertibles] a third follows, then from
either of them together with the contradictory of the conclusion the
contradictory of the other follows (Apuleius
Int. 209.9–14).

This is an inference rule of the kind today called antilogism. The
third thema, in one formulation, was:

When from two [assertibles] a third follows, and from the
one that follows [i.e. the third] together with another,
external assumption, another follows, then this other follows from the
first two and the externally co-assumed one (Simplicius Cael.
237.2–4).

This is an inference rule of the kind today called cut-rule. It is
used to reduce chain-syllogisms. The second and fourth
themata were also cut-rules, and reconstructions of them can
be provided, since we know what arguments they together with the third
thema were thought to reduce, and we have some of the
arguments said to be reducible by the second thema. A
possible reconstruction of the second thema is:

When from two assertibles a third follows, and from the third and
one (or both) of the two another follows, then this other follows from
the first two.

A possible reconstruction of the fourth thema is:

When from two assertibles a third follows, and from the third and
one (or both) of the two and one (or more) external assertible(s)
another follows, then this other follows from the first two and the
external(s). (Cf. Bobzien 1996.)

A Stoic reduction shows the formal validity of an argument by applying
to it the themata in one or more steps in such a way that all
resultant arguments are indemonstrables. This can be done either with
the arguments or their modes (S. E. M
8.230–8). For instance, the argument mode

If the 1st and the 2nd, the 3rd.
But not the 3rd.
Moreover, the 1st.
Therefore not: the 2nd.

can be reduced by the third thema to (the modes of) a second
and a third indemonstrable as follows:

When from two assertibles (‘If the 1st and the
2nd, the 3rd’ and ‘But not the
3rd’) a third follows (‘Not: both the
1st and the 2nd’—this follows by a
second indemonstrable) and from the third and an external one
(‘The 1st’) another follows (‘Not: the
2nd’—this follows by a third indemonstrable),
then this other (‘Not: the 2nd’) also follows
from the two assertibles and the external one.

The second thema reduced, among others, arguments with the
following modes (Alex. Aphr. An. Pr. 164.27–31):

Either the 1st or not the 1st.
But the 1st.
Therefore the 1st.

If the 1st, if the 1st, the 2nd.
But the 1st.
Therefore the 2nd.

The Peripatetics chided the Stoics for allowing such useless
arguments. In agreement with contemporary logic, the Stoics insisted
that, if the arguments can be reduced, they are valid.

The four themata can be used repeatedly and in any
combination in a reduction. Thus propositional arguments of
indeterminate length and complexity can be reduced. Stoic syllogistic
has been formalized, and it has been shown that the Stoic deductive
system shows strong similarities with relevance logical systems like
those by Storrs McCall. Like Aristotle, the Stoics aimed at proving
non-evident formally valid arguments by reducing them by
means of accepted inference rules to evidently valid
arguments. Thus, although their logic is a propositional
logic, they did not intend to provide a system that allows for the
deduction of all propositional-logical truths, but rather a system of
valid propositional-logical arguments with at least two premises and a
conclusion. Nonetheless, we have evidence that the Stoics expressly
recognized many simple logical truths. For example, they accepted the
following logical principles: the principle of double negation,
stating that a double negation (‘not: not: p’) is
equivalent to the assertible that is doubly negated (i.e. p)
(D. L. 7.69); the principle that any conditional formed by using the
same assertible as antecedent and as consequent (‘if p,
p’) is true (S. E. M 8.281, 466); the principle
that any two-place disjunctions formed by using contradictory
disjuncts (‘either p or not: p’) is true
(S. E. M 8.282, 467); and the principle of contraposition,
that if ‘if p, q’ then ‘if not:
q, not: p’ (D. L. 7.194, Philodemus
Sign., PHerc. 1065, XI.26–XII.14).

The Stoics recognized the importance of both the Liar and the Sorites
paradoxes (Cicero Acad. 2.95–8, Plut.
Comm.Not. 1059D–E, Chrys. Log. Zet. col.IX).
Chrysippus may have tried to solve the Liar as follows: there is an
ineliminable ambiguity in the Liar sentence (‘I am speaking
falsely’, uttered in isolation) between the assertibles (i)
‘I falsely say I speak falsely’ and (ii) ‘I
am speaking falsely’ (i.e. I am doing what I'm saying,
viz. speaking falsely), of which, at any time the Liar sentence is
uttered, precisely one is true, but it is arbitrary which one. (i)
entails (iii) ‘I am speaking truly’ and is
incompatible with (ii) and with (iv) ‘I truly say I speak
falsely’. (ii) entails (iv) and is incompatible with (i) and
(iii). Thus bivalence is preserved (cf. Cavini 1993). Chrysippus'
stand on the Sorites seems to have been that vague borderline
sentences uttered in the context of a Sorites series have no
assertibles corresponding to them, and that it is obscure to us where
the borderline cases start, so that it is rational for us to stop
answering while still on safe ground (i.e. before we might begin to
make utterances with no assertible corresponding to them). The latter
remark suggests Chrysippus was aware of the problem of higher order
vagueness. Again, bivalence of assertibles is preserved (cf. Bobzien
2002). The Stoics also discussed various other well-known
paradoxes. In particular, for the paradoxes of presupposition, known
in antiquity as the Horned One, they produced a Russellian-type
solution based on a hidden scope ambiguity of negation (cf. Bobzien
2012)

Epicurus (late 4th–early 3rd c. BCE) and
the Epicureans are said to have rejected logic as an unnecessary
discipline (D. L. 10.31, Usener 257). This notwithstanding, several
aspects of their philosophy forced or prompted them to take a stand on
some issues in philosophical logic. (1) Language meaning and
definition: The Epicureans held that natural languages came into
existence not by stipulation of word meanings but as the result of the
innate capacities of humans for using signs and articulating sounds
and of human social interaction (D. L. 10.75–6); that language is
learnt in context (Lucretius 5.1028ff); and that linguistic
expressions of natural languages are clearer and more conspicuous than
their definitions; even that definitions would destroy their
conspicuousness (Usener 258, 243); and that philosophers hence should
use ordinary language rather than introduce technical expressions
(Epicurus On Nature 28). (2) Truth-bearers: the
Epicureans deny the existence of incorporeal meanings, such as Stoic
sayables. Their truth-bearers are linguistic items, more precisely,
utterances (phônai) (S. E. M 8.13, 258; Usener
259, 265). Truth consists in the correspondence of things and
utterances, falsehood in a lack of such correspondence
(S. E. M 8.9, Usener 244), although the details are obscure
here. (3) Excluded middle: with utterances as truth-bearers,
the Epicureans face the question what the truth-values of future
contingents are. Two views are recorded. One is the denial of the
Principle of Excluded Middle (‘p or not
p’) for future contingents (Usener 376, Cicero
Acad. 2.97, Cicero Fat. 37). The other, more
interesting, one leaves the Excluded Middle intact for all utterances,
but holds that, in the case of future contingents, the component
utterances ‘p’ and ‘not p’
are neither true nor false (Cicero Fat. 37), but, it seems,
indefinite. This could be regarded as an anticipation of
supervaluationism. (4) Induction: Inductive logic was
comparatively little developed in antiquity. Aristotle discusses
arguments from the particular to the universal
(epagôgê) in the Topics and
Posterior Analytics but does not provide a theory of
them. Some later Epicureans developed a theory of inductive inference
which bases the inference on empirical observation that certain
properties concur without exception (Philodemus De
Signis).

Very little is known about the development of logic from c. 100 BCE to
c. 250 CE. It is unclear when Peripatetics and Stoics began taking
notice of each others' logical achievements. At some point during that
period, the terminological distinction between ‘categorical
syllogisms’, used for Aristotelian syllogisms, and
‘hypothetical syllogisms’, used not only for those
introduced by Theophrastus and Eudemus, but also for the Stoic
propositional-logical syllogisms, gained a foothold. In the first
century BCE, the Peripatetics Ariston of Alexandria and Boethus of
Sidon wrote about syllogistic. Ariston is said to have introduced the
so-called ‘subaltern’ syllogisms (Barbari, Celaront,
Cesaro, Camestrop and Camenop) into Aristotelian syllogistic (Apuleius
Int. 213.5–10), i.e. the syllogisms one gains by
applying the subalternation rules (that were acknowledged by Aristotle
in his Topics)

From ‘A holds of every B’ infer
‘A holds of some B’

From ‘A holds of no B’ infer
‘A does not hold of some B’

to the conclusions of the relevant syllogisms. Boethus suggested
substantial modifications to Aristotle's theories: he claimed that all
categorical syllogisms are complete, and that hypothetical syllogistic
is prior to categorical (Galen Inst. Log. 7.2), although we
are not told what this priority was thought to consist in. The Stoic
Posidonius (c. 135–c. 51 BCE) defended the possibility of
logical or mathematical deduction against the Epicureans and discussed
some syllogisms he called ‘conclusive by the force of an
axiom’, which apparently included arguments of the type
‘As the 1st is to the 2nd, so the
3rd is to the 4th; the ratio of the
1st to the 2nd is double; therefore the ratio of
the 3rd to the 4th is double’, which was
considered conclusive by the force of the axiom ‘things which
are in general of the same ratio, are also of the same particular
ratio’ (Galen Inst. Log. 18.8). At least two Stoics in
this period wrote a work on Aristotle's Categories. From his
writings we know that Cicero (1st c. BCE) was knowledgeable
about both Peripatetic and Stoic logic; and Epictetus' discourses
(late 1st–early 2nd c. CE) prove that he
was acquainted with some of the more taxing parts of Chrysippus'
logic. In all likelihood, there existed at least a few creative
logicians in this period, but we do not know who they were or what
they created.

The next logician of rank, if of lower rank, of whom we have
sufficient evidence to speak is Galen (129–199 or 216 CE), who
achieved greater fame as a physician. He studied logic with both
Peripatetic and Stoic teachers, and recommended availing oneself of
parts of either doctrine, as long as it could be used for scientific
demonstration. He composed commentaries on logical works by Aristotle,
Theophrastus, Eudemus and Chrysippus, as well as treatises on various
logical problems and a major work entitled On
Demonstration. All these are lost, except for some information in
later texts, but his Introduction to Logic has come down to
us almost in full. In On Demonstration, Galen developed,
among other things, a theory of compound categorical syllogisms with
four terms, which fall into four figures, but we do not know the
details. He also introduced the so-called relational syllogisms,
examples of which are ‘A is equal to B,
B is equal to C; therefore A is equal to
C’ and ‘Dio owns half as much as Theo; Theo owns
half as much as Philo. Therefore Dio owns a quarter of what Philo
owns’ (Galen Inst. Log, 17–18). All the relational
syllogisms Galen mentions have in common that they are not reducible
in either Aristotle's or the Stoic syllogistic, but it is difficult to
find further formal characteristics that unite them. In general, in
his Introduction to Logic Galen merges Aristotelian
Syllogistic with a strongly Peripatetic reinterpretation of Stoic
propositional logic. This becomes apparent in particular in Galen's
emphatic denial that truth-preservation is sufficient for the validity
or syllogismhood of an argument, and his insistence that, instead,
knowledge-introduction or knowledge-extension is a necessary condition
for something to count as a
syllogism.[5]

The second ancient introduction to logic that has survived is
Apuleius' (2nd cent. CE) De Interpretatione. This
Latin text, too, displays knowledge of Stoic and Peripatetic logic; it
contains the first full presentation of the square of opposition,
which illustrates the logical relations between categorical sentences
by diagram. The Platonist Alcinous (2nd cent. CE), in his
Handbook of Platonism chapter 5, is witness to the emergence
of a specifically Platonist logic, constructed on the Platonic notions
and procedures of division, definition, analysis and hypothesis, but
there is little that would make a logician's heart beat
faster. At some time between the 3rd and 6th century
CE Stoic logic faded into oblivion, to be resurrected only in the
20th century, in the wake of the (re)-discovery of
propositional logic.

The surviving, often voluminous, Greek commentaries on Aristotle's
logical works by Alexander of Aphrodisias (fl. c. 200 CE), Porphyry
(234–c. 305), Ammonius Hermeiou (5th century),
Philoponus (c. 500) and Simplicius (6th century) and the
Latin ones by Boethius (c. 480–524) are mainly important for
preserving alternative interpretations of Aristotle's logic and as
sources for lost Peripatetic and Stoic works. They also allow us to
trace the gradual development from a Peripatetic exegesis of
Aristotle’s Organon to a more eclectic logic that resulted
from the absorption and inclusion of elements not just from Stoic and
Platonist theories but also from mathematics and rhetoric. Two of the
commentators in particular deserve special mention in their own right:
Porphyry, for writing the Isagoge or Introduction
(i.e. to Aristotle's Categories), in which he discusses the
five notions of genus, species, differentia, property and accident as
basic notions one needs to know to understand the
Categories. For centuries, the Isagoge was the first
logic text a student would tackle, and Porphyry's five predicables
(which differ from Aristotle's four) formed the basis for the medieval
doctrine of the quinque voces. The second is Boethius. In
addition to commentaries, he wrote a number of logical treatises,
mostly simple explications of Aristotelian logic, but also two very
interesting ones: (i) His On Topical Differentiae bears
witness to the elaborated system of topical arguments that logicians
of later antiquity had developed from Aristotle's Topics
under the influence of the needs of Roman lawyers. (ii) His On
Hypothetical Syllogisms systematically presents wholly
hypothetical and mixed hypothetical syllogisms as they are known from
the early Peripatetics; it may be derived from Porphyry. Boethius'
insistence that the negation of ‘If it is A, it is
B’ is ‘If it is A, it is not
B’ suggests a suppositional understanding of the
conditional, a view for which there is also some evidence in Ammonius,
but that is not attested for earlier logicians. Historically, Boethius
is most important because he translated all of Aristotle's
Organon into Latin, making these texts (except the
Posterior Analytics) available to philosophers of the
medieval period.